Homework Help: Equations of Motion for a mass on a circular track with a linear spring

1. Sep 11, 2012

tlonster

1. The problem statement, all variables and given/known data

A collar of mass m slides without friction along a circular track of radius R as
shown in THE ATTACHMENT. Attached to the collar is a linear spring with spring constant K and
unstretched length zero. The spring is attached at the fixed point A located a distance
2R from the center of the circle. Assume gravity acts down and determine (a) the differential equation of motion for the collar in terms the angle θ and (b) the reaction force exerted by the track on the collaras a function of the angle θ.

2. Relevant equations

The transport theorem

F = ma

3. The attempt at a solution

I have worked through this problem and am not confident in my answer. A second look would be much appreciated. Here's my answer:

I created reference frame E: {er, eθ, ez} at mass m such that it's position is

r = Rer

By using the transport theorem for velocity and acceleration as seen by the intertial frame I came up with:

v(m/I) = 0 + θ(dot)R[eθ]
a(m/I) = θ(double dot)R[eθ] - (θ(dot))^2R[er]

Next the forces acting on my mass are gravity, normal, and linear spring

∴Ʃ F = mg[JI] + N[er] + K(3R-0)[-er + JI]

where JI = sinθ[er] + cosθ[eθ]

when I substitute JI into my ƩF equation, set equal to ma(m/I), and drop the unit vectors, I get these two equations:

1. mgsinθ + N -3KR + 3KRsinθ = -m(θ(dot))^2R
2. mgcosθ + 3KRcosθ = mθ(double dot)R

For my equation of motion in terms of θ, I only need to use the second equation, correct? Where θ(double dot) = (mgcosθ + 3KRcosθ)/(mR)

and for the normal force (N) I just use the first equation and move everything to one side for N = -mgsinθ +3KR - 3KRsinθ -m(θ(dot))^2R

Can someone please take a look at this? I feel as though my spring force
Fs = K(3R-0)[-er + JI] is not correct, but I don't know what else it should be. Any help would be appreciated.

Thank you.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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